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arXiv:1605.00655 · Physical review. B./Physical review. B

Many-body localization beyond eigenstates in all dimensions

Anushya Chandran, Arijeet Pal, Chris R. Laumann, Antonello Scardicchio

Published Oct 7, 2016• 75 citations
arXivHTMLPDF

Abstract

Isolated quantum systems with quenched randomness exhibit many-body localization (MBL), wherein they do not reach local thermal equilibrium even when highly excited above their ground states. It is widely believed that individual eigenstates capture this breakdown of thermalization at finite size. We show that this belief is false in general and that a MBL system can exhibit the eigenstate properties of a thermalizing system. We propose that localized approximately conserved operators $({\mathrm{l}}^{*}\text{-bits})$ underlie localization in such systems. In dimensions $d>1$, we further argue that the existing MBL phenomenology is unstable to boundary effects and gives way to ${\mathrm{l}}^{*}\text{-bits}$. Physical consequences of ${\mathrm{l}}^{*}\text{-bits}$ include the possibility of an eigenstate phase transition within the MBL phase unrelated to the dynamical transition in $d=1$ and thermal eigenstates at all parameters in $d>1$. Near-term experiments in ultracold atomic systems and numerics can probe the dynamics generated by boundary layers and emergence of ${\mathrm{l}}^{*}\text{-bits}$.

DOI
10.1103/physrevb.94.144203

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Isolated quantum systems with quenched randomness exhibit many-body localization (MBL), wherein they do not reach local thermal equilibrium even when highly excited above their ground states. It is widely believed that individual eigenstates capture this breakdown of thermalization at finite size. We show that this belief is false in general and that a MBL system can exhibit the eigenstate properties of a thermalizing system. We propose that localized approximately conserved operators (l∗-bits)({\mathrm{l}}^{*}\text{-bits})(l∗-bits) underlie localization in such systems. In dimensions d>1d>1d>1, we further argue that the existing MBL phenomenology is unstable to boundary effects and gives way to l∗-bits{\mathrm{l}}^{*}\text{-bits}l∗-bits. Physical consequences of l∗-bits{\mathrm{l}}^{*}\text{-bits}l∗-bits include the possibility of an eigenstate phase transition within the MBL phase unrelated to the dynamical transition in d=1d=1d=1 and thermal eigenstates at all parameters in d>1d>1d>1. Near-term experiments in ultracold atomic systems and numerics can probe the dynamics generated by boundary layers and emergence of l∗-bits{\mathrm{l}}^{*}\text{-bits}l∗-bits.
Many-body localized (MBL) systems lie outside the framework of statistical mechanics, as they fail to equilibrate under their own quantum dynamics. Even basic features of MBL systems, such as their stability to thermal inclusions and the nature of the dynamical transition to thermalizing behaviour, remain poorly understood. We study a simple central spin model to address these questions: a two-level system interacting with strength J with N ≫1 localized bits subject to random fields. On increasing J , the system transitions from an MBL to a delocalized phase on the vanishing scale J c ( N )∼1/ N , up to logarithmic corrections. In the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either ‘on’ (strongly entangled) or ‘off’ (weakly entangled) in eigenstates. The clusters of ‘on’ bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogeneous discontinuous transition out of the localized phase in single-site observables. We obtain these results by perturbative mapping to bond percolation on the hypercube at small J and by numerical exact diagonalization of the full many-body system. Our results support the arguments that the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions d that are high but finite. This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.
Despite the exponentially large amount of information required in the quantum description of many-body systems, finite size numerical simulations have paved the way for recent progress in the understanding of quantum phases of matter both in and out-of-equilibrium. \n \nIn this spirit, using exact diagonalization for a small number of spins ~ 20, first, we investigate the fate of both ergodic quantum many-body systems that thermalize at long times and systems with quenched disorder that exhibit many-body localization (MBL) and fail to thermalize (due to an extensive set of local conserved degrees of freedom) under periodic driving. Ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local driving, in which the time dependence of the Hamiltonian is restricted to a few contiguous degrees of freedom, and global driving. On the other hand, MBL systems remain localized at finite energy density for local driving, while in the case of global driving these systems can either heat up to infinite temperature or remain localized depending on the parameters of the drive. Underlying the latter is the emergence of an effective time independent MBL Hamiltonian describing the long time dynamics of the system. These numerical predictions have recently been verified experimentally in a one dimensional quasi-disordered optical lattice of interacting fermions wherein unitary time evolution can be experimentally probed. \n \nSubsequently, we study the stability of MBL systems with respect to another perturbation, in this case by thermal inclusions, which remains poorly understood. We consider a simple model to address this question: a two level system interacting with 𝑁 localized bits subject to random fields. In the large 𝑁 limit, the localized bits are thermalized for any nonzero ratio of interactions to disorder strength. Moreover, at finite 𝑁, in the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either "on" (strongly entangled) or "off" (weakly entangled) in eigenstates. The clusters of "on" bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogenous discontinuous transition out of the localized phase in single site observables. These results also imply the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions 𝑑 that are high but finite. \n \nThe second part of this thesis is motivated by the recent breakthroughs of artificial intelligence in domains such as computer vision involving highly dimensional data of complexity not unlike the one found in the description of many-body systems. Using Monte Carlo simulations to generate synthetic data, we find that kernel methods, a class of machine learning algorithms, can learn the mathematical \nform of physical discriminators associated with phase transitions, such as order parameters and Hamiltonian constraints, for a set of two dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. \n \nOur work shows that the combination of numerical simulations with the ability of machine learning algorithms for automatic feature detection hold promise to become standard tools in the exploration and characterization of different phases of matter in condensed matter and other many-body systems.
We review some recent developments in the statistical mechanics of isolated quantum systems. We provide a brief introduction to quantum thermalization, paying particular attention to the eigenstate thermalization hypothesis (ETH) and the resulting single-eigenstate statistical mechanics. We then focus on a class of systems that fail to quantum thermalize and whose eigenstates violate the ETH: These are the many-body Anderson-localized systems; their long-time properties are not captured by the conventional ensembles of quantum statistical mechanics. These systems can forever locally remember information about their local initial conditions and are thus of interest for possibilities of storing quantum information. We discuss key features of many-body localization (MBL) and review a phenomenology of the MBL phase. Single-eigenstate statistical mechanics within the MBL phase reveal dynamically stable ordered phases, and phase transitions among them, that are invisible to equilibrium statistical mechanics and can occur at high energy and low spatial dimensionality, where equilibrium ordering is forbidden.
The nature of the many-body localization (MBL) transition and even the existence of the MBL phase in random many-body quantum systems have been actively debated in recent years. In spatial dimension d>1d>1d>1, there is some consensus that the MBL phase is unstable to rare thermal inclusions that can lead to an avalanche that thermalizes the whole system. In this note, we explore the possibility of MBL in quasiperiodic systems in dimension d>1d>1d>1. We argue that (i) the MBL phase is stable against ``avalanches'' at strong enough quasiperiodic modulations for d=2d=2d=2, and (ii) the possibility of an avalanche strongly constrains the finite-size scaling behavior of the MBL transition. We present a suggestive construction that MBL is unstable for $d\ensuremath{\ge}3$.
Abstract Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBL regime give rise to an MBL phase , in which the thermalization does not occur even in the asymptotic limit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.
Disordered quantum many-body systems pose one of the central challenges in condensed matter physics and quantum information science, as their dynamics are generally intractable for classical computation. Many-body localization (MBL), hypothesized to evade thermalization indefinitely under strong disorder, exemplifies this difficulty. Here, we study the stability of MBL in two dimensions using ultracold atoms in optical lattices with variable system sizes up to 24×2424\times 2424×24 sites, well beyond the classically simulable regime. Using the imbalance as a probe, we trace the long-time dynamics under two distinctive disorder potentials: quasiperiodic and random disorder. For random disorder, the MBL crossover point shifts to higher disorder strength with increasing system size, consistent with the avalanche scenario. In contrast, with quasiperiodic disorder, we observe no clear system size dependence, suggesting possible stability of MBL in two dimensions.